A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators

TL;DR

This paper completes the analysis of Gevrey well-posedness for hyperbolic operators with non-effectively hyperbolic double characteristics, extending previous results to cases with arbitrary codimension of the characteristic manifold.

Contribution

It removes the restriction on the codimension of the characteristic manifold, completing the characterization of well-posedness for these operators.

Findings

Gevrey well-posedness thresholds are established for a broader class of hyperbolic operators.

The results confirm the optimality of the thresholds in the extended setting.

The analysis links the Hamilton map properties to well-posedness in Gevrey classes.

Abstract

For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In our previous work, we showed that if the Hamilton map has a Jordan block of size $4$ on the double characteristic manifold $\Sigma$ of codimension $3$, then the Cauchy problem for $P$ is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and that this result is optimal. Moreover, if there are no bicharacterisitcs tangent to $\Sigma$, then the Cauchy problem is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and this result is also optimal. In the present paper, we remove the restriction on the codimension of $\Sigma$, thereby completing the result.